Cosine and Sine graphs are very similar. So similar, in fact, that the Cosine graph is exactly the Sine graph with a phase shift of \(\frac{\pi}{{2}}\) (aka, 90 degrees). If you look at the Unit Circle, that should make sense; the \(x\) and \(y\) axis represent Cosine and Sine, after all, and they are just a \(\frac{\pi}{{2}}\) rotation of one another.

Thus, when it comes to graphing the Cosine, all the same reasoning applies! There is really only one critical difference to consider when graphing the Cosine function:

Cosine will begin at either a maximum or a minimum, whereas the Sine function begins at the midline.

Let's look at the basic \(y=\cos(\theta)\) graph to see what we mean:

Notice that the graph begins at \((0,1)\)! 

Similar to the Sine function, in general:

\[y=A\cos\left(B(\theta-\omega)\right)+C\]

has the following properties (differences from the Sine function will be in red):

• Domain: All real numbers
• Amplitude: \(|A|\)
• Midline/Average Value: \(y=C\)
• Range: \(\left[C-|A|,C+|A|\right]\)
• Phase Shift: \(\omega\)
• Frequency: \(f=\frac{{B}}{2\pi}\)
• Period: \(P=\frac{2\pi}{{B}}\)
• Max/Min: Can be found every half-period, offset by the phase shift. So, each max/min(there would be infinitely many) could be written with the formula \(\frac{{nP}}{{2}} +\omega\), where \(n\) is any integer and \(P\) is the period.
• The Midline intersections will be found every quarter-period (in between the max/mins)

 We will explore this in greater detail in the following examples.